<p>In this paper, we investigate random operators on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathbb {Z}}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> with Hölder continuously distributed potentials and the long-range hopping. The hopping amplitude decays with the inter-particle distance <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Vert \varvec{x}\Vert $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">‖</mo> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">‖</mo> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$e^{-\log ^{\rho }(\Vert \varvec{x}\Vert +1)}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msup> <mo>log</mo> <mi>ρ</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">‖</mo> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">‖</mo> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\rho >1,\varvec{x}\in {\mathbb {Z}}^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo>></mo> <mn>1</mn> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. By employing the multi-scale analysis (MSA) technique, we prove that for large disorder, the random operators have pure point spectrum with localized eigenfunctions whose decay rate is the same as the hopping term. This gives a partial answer to a conjecture of Yeung and Oono [<i>Europhys. Lett.</i> 4(9), (1987): 1061-1065].</p>

Localization for Random Operators on ${\mathbb {Z}}^d$ with the Long-Range Hopping

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2025年6月13日

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Yunfeng Shi,
Li Wen,
Dongfeng Yan

摘要

In this paper, we investigate random operators on ${\mathbb {Z}}^d$ $Z^{d}$ with Hölder continuously distributed potentials and the long-range hopping. The hopping amplitude decays with the inter-particle distance $\Vert \varvec{x}\Vert $ $‖ x ‖$ as $e^{-\log ^{\rho }(\Vert \varvec{x}\Vert +1)}$ $e^{- {log}^{ρ} (‖ x ‖ + 1)}$ with $\rho >1,\varvec{x}\in {\mathbb {Z}}^d$ $ρ > 1, x \in Z^{d}$ . By employing the multi-scale analysis (MSA) technique, we prove that for large disorder, the random operators have pure point spectrum with localized eigenfunctions whose decay rate is the same as the hopping term. This gives a partial answer to a conjecture of Yeung and Oono [Europhys. Lett. 4(9), (1987): 1061-1065].

Localization for Random Operators on \({\mathbb {Z}}^d\) with the Long-Range Hopping

摘要